Reiter's condition \(P_1\) and approximate identities for polynomial hypergroups
Let \(K\) be a commutative hypergroup, \(\mu \) its Haar measure. In order to give a necessary and sufficient condition for the existence of bounded approximate identities in the maximal ideals of the Banach algebra \( L^{1}(K,\mu )\),\ in [\textit{F. Filbir}, ``Existence of approximate identities in maximal ideals of hypergroup-algebras'', in: Special Functions. Proc. Int. Workshop Hong Kong 1999, 88--97] a ``modified '' Reiter condition \(\mathcal{P}_{1} \) has been introduced. Continuing this work, in the present paper it is shown that the modified Reiter condition \(\mathcal{P}_{1}\)\ is satisfied in \( \alpha \in \widehat{K}\) (or, equivalently \(I(\alpha )\), the ideal generated by the character \(\alpha \), has a bounded approximate identity) if and only if there exists \(m_{\alpha }\in (L^{\infty }(K,\mu ))^{\ast }\) with similar properties to an invariant mean: \(m_{\alpha }(\alpha )=1\), \( \parallel m_{\alpha }\parallel =1\), \(m_{\alpha }(T_{y}f)=\alpha (y)\) \( m_{\alpha }(f)\), \(\forall f\in L^{\infty }(K,\mu )\), \(y\in K\). As is mentioned in the paper, the existence of generalized means for characters of \(K\) could be the basic tool to carry on harmonic analysis for example on the bidual of \(L^{1}(K,\mu )\) or to study some important classes of \ functions on \(K\) such as almost periodic, weakly almost periodic or uniformly continuous functions more thoroughly. In the final part of the paper, the obtained results are applied to polynomial hypergroups (especially to the ones generated by Jacobi polynomials).
scientific article; zbMATH DE number 2132192